In statistics and

econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8� ...

, extremum estimators are a wide

class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently ...

estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...

s for parametric models that are calculated through maximization (or minimization) of a certain

objective function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...

, which depends on the data. The general theory of extremum estimators was developed by .

Definition

An estimator

\scriptstyle\hat\theta

is called an extremum estimator, if there is an ''objective function''

\scriptstyle\hat_n

such that :

\hat\theta = \underset\ \widehat_n(\theta),

where Θ is the

parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for ...

. Sometimes a slightly weaker definition is given: :

\widehat Q_n(\hat\theta) \geq  \max_\,\widehat Q_n(\theta) - o_p(1),

where ''o''_''p''(1) is the variable converging in probability to zero. With this modification

\scriptstyle\hat\theta

doesn't have to be the exact maximizer of the objective function, just be sufficiently close to it. The theory of extremum estimators does not specify what the objective function should be. There are various types of objective functions suitable for different models, and this framework allows us to analyse the theoretical properties of such estimators from a unified perspective. The theory only specifies the properties that the objective function has to possess, and so selecting a particular objective function only requires verifying that those properties are satisfied.

Consistency

If the parameter space Θ is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...

and there is a ''limiting function'' ''Q''₀(''θ'') such that:

\scriptstyle\hat_n(\theta)

converges to ''Q''₀(''θ'') in probability uniformly over Θ, and the function ''Q''₀(''θ'') is

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

and has a unique maximum at ''θ'' = ''θ''₀. If these conditions are satisfied then

\scriptstyle\hat\theta

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

for ''θ''₀. The uniform convergence in probability of

\scriptstyle\hat_n(\theta)

means that :

\sup_ \big,  \hat_n(\theta) - Q_0(\theta) \big,  \ \xrightarrow\ 0.

The requirement for Θ to be compact can be replaced with a weaker assumption that the maximum of ''Q''₀ was well-separated, that is there should not exist any points ''θ'' that are distant from ''θ''₀ but such that ''Q''₀(''θ'') were close to ''Q''₀(''θ''₀). Formally, it means that for any sequence such that , it should be true that .

Asymptotic normality

Assuming that consistency has been established and the derivatives of the sample

Q_

satisfy some other conditions, the extremum estimator converges to an asymptotically Normal distribution.

Examples

Notes

References